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Cooperation with: P. Dai Pra (Università degli Studi di Padova, Italy), D. Dereudre (École Polytechnique, Palaiseau, France), M. Sortais (Berliner Graduiertenkolleg ,,Stochastische Prozesse und Probabilistische Analysis`` (Graduate College ``Stochastic Processes and Probabilistic Analysis'')), M. Thieullen (Université Paris VI ``Pierre et Marie Curie'', France), L. Zambotti (Scuola Normale Superiore di Pisa, Italy, and Technische Universität Berlin), H. Zessin (Universität Bielefeld)
Supported by: Alexander von HumboldtStiftung (Alexander von Humboldt Foundation): Fellowship
Description:
In the following works, we are interested in the analysis of several types of interactive diffusions modeling phenomena coming either from Statistical Physics or from Population Dynamics. The underlying idea is to transpose important concepts and tools from Statistical Mechanics like Gibbs equilibrium measures, entropy, spacetime limit, to mathematical objects like diffusions, or Brownian semimartingales. Working on path space, we also obtain a better understanding of the behavior of such diffusions.
Together with M. Thieullen we study the class of all probabilities on the path space which have the same bridges as the following valued reference Brownian diffusion denoted by P_{b} and law of the solution of the stochastic differential system:
where the drift is a regular function on .
Equation (1) is a perturbation of the duality equation
satisfied by Brownian bridges, duality between the
Malliavin derivation operator and the stochastic integral. The
perturbation terms (second and third terms in the RHS of equation
(1) ) are to be compared with the Malliavin derivatives of
the
Hamiltonian function associated to Gibbs measures. The main difference from
the onedimensional situation studied before in [8]
comes from the new last term in (1), the stochastic integral
of
the reciprocal characteristic G w.r.t. the
coordinate process. This term vanishes if and only if the drift b of the
reference
Brownian diffusion is a gradient. In [8] this term was
identically zero since each regular function is a gradient in dimension
N=1.
One finds in [9] several applications of this characterization
of
reciprocal processes. Among others, the authors prove a generalization of
the famous Kolmogorov
theorem about reversible diffusion:
the existence of a reversible law in the reciprocal class of a
Brownian diffusion with drift b can only
occur if b is a gradient. In collaboration with L. Zambotti, they also
study an application of the duality equation (1) in the
singular case of Bessel processes. There the difficulty consists in
generalizing the reciprocal characteristics F and G associated to the degenerate
drift function .
The infinitedimensional
situation, when the index i gets continuous () and the reference
process is solution of a stochastic partial
differential equation, is the next step in the study of such topics. It will
be the subject
of a forthcoming paper.
With D. Dereudre [3] we study
Gibbsian properties on the path level of continuous systems of
infinitedimensional
interactive Brownian diffusion. More precisely we consider the law of the solution of
the following stochastic differential system with values in :
(2) 
In collaboration with P. Dai Pra and H. Zessin [5], we give a
Gibbsian characterization for the stationary law of the interacting
diffusion process
defined on the lattice and solution of the following type
of
stochastic differential system
(3) 
In [4], based on the Gibbsian characterization shown in [5], a weak existence result for the solution of (3) is proved. The authors use a spacetime clusterexpansion method, which is powerful when the coupling parameter is sufficiently small. As conclusion, the Gibbsian approach to study infinitedimensional processes seems to be very effective in situations where the stochastic calculus can not give an answer, like for example in the question of existence of solution for the nonMarkovian equation (3).
Consider now the following model of Statistical Physics in random
medium:
Langevin dynamics for a ferromagnetic
system submitted to a disordered external Bernoulli magnetic field:
for ,
(4) 
In [7] the authors use the clusterexpansion method to show that the spacetime correlation functions associated to Y decay exponentially fast in the hightemperature regime. In particular, they prove that for small enough, the infinitedimensional process Y(t) is ergodic and the velocity of the convergence is exponential. The specific difficulty of this model (compared to the system (3)) comes from the fact that the nonMarkovian interaction (last term in the RHS of (5)) is no more local in time: it depends on the values of Y on the full time interval [0,t].
The process studied in [10] arises as diffusion
limit of branching particle systems. It is called
catalytic superBrownian motion, in the sense that the branching procedure
in the approximation depends strongly on a singular medium called
catalyst. It is modeled as solution of the following stochastic partial
differential equation:
(6) 
References:



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